Mathematics

# Evaluate the following integral$\int { \cfrac { \sin { 2x } }{ a\cos ^{ 2 }{ x } +b\sin ^{ 2 }{ x } } } dx$

##### SOLUTION
$\displaystyle\int{\dfrac{\sin{2x} \, dx}{a{\cos}^{2}{x}+b{\sin}^{2}{x}}}$

Let
$t=a{\cos}^{2}{x}+b{\sin}^{2}{x}\Rightarrow\,dt=\left(-2a\cos{x}\sin{x}+2b\sin{x}\cos{x}\right)dx$

$\Rightarrow\,\dfrac{dt}{b-a}=2\sin{x}\cos{x}dx$

$\Rightarrow\,\dfrac{dt}{b-a}=\sin{2x}dx$

$\displaystyle\int{\dfrac{\sin{2x}dx}{a{\cos}^{2}{x}+b{\sin}^{2}{x}}}$

$=\dfrac{1}{b-a}\displaystyle\int{\dfrac{dt}{t}}$

$=\dfrac{1}{b-a}\log{\left|t\right|}+c$

$=\dfrac{1}{b-a}\log{\left|a{\cos}^{2}{x}+b{\sin}^{2}{x}\right|}+c$ where $t=a{\cos}^{2}{x}+b{\sin}^{2}{x}$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Single Correct Medium
If $\displaystyle \frac{(1+\mathrm{x})(1+2\mathrm{x})(1+3\mathrm{x})}{(1-\mathrm{x})(1-2\mathrm{x})(1-3\mathrm{x})}=\mathrm{K}+$ $\displaystyle \frac{\mathrm{A}}{1-\mathrm{x}}+\frac{\mathrm{B}}{1-2\mathrm{x}}+\frac{\mathrm{C}}{1-3\mathrm{x}}$, then which of the following is correct
• A. $\mathrm{K}=6$
• B. $\mathrm{B}=30$
• C. $\mathrm{C}=-20$
• D. $\mathrm{A}=12$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate: $- 2 \int \cos 2 \theta \sin 2\theta d \theta$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
If $\displaystyle f\left ( x \right )=\int_{-1}^{1}\frac{\sin x}{1+t^{2}}dt$ then $\displaystyle {f}'\left ( \frac{\pi }{3} \right )$ is
• A. nonexistent
• B. $\displaystyle \pi \sqrt{3/4}$
• C. none of these
• D. $\displaystyle \pi /4$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
$\displaystyle\int \dfrac{ln \left(\dfrac{x-1}{x+1}\right)}{x^2-1}dx$ is equal to?

$\int_{}^{} {\frac{{ - 1}}{{\sqrt {1 - {x^2}} }}dx}$